Multi-patterning graph reduction and checking flow method

ABSTRACT

A method of generating a plurality of photomasks includes generating a circuit graph. The circuit graph comprises a plurality of vertices and a plurality of edges. Each of the plurality of vertices is representative of one of a plurality of conductive lines. The plurality of edges are representative of a spacing between the conductive lines less than an acceptable minimum distance. K n+1  graph comprising a first set of vertices selected from the plurality of vertices connected in series by a first set of edges selected from the plurality of edges and having at least one non-series edge connection between a first vertex and a second vertex selected from the first set of vertices is reduced by merging a third vertex into a fourth vertex selected from the first set of the plurality of vertices. An n-pattern conflict check is performed and the photomasks generated based on the result.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 15/255,489, filed Sep. 2, 2016, and entitled “Multi-Patterning Graph Reduction and Checking Flow Method,” the disclosure of which is incorporated herein by reference in its entirety.

FIELD

This disclosure relates to semiconductor fabrication generally, and more specifically to multi-patterning semiconductor mask checking.

BACKGROUND

In semiconductor integrated circuit (IC) fabrication processes, the resolution of a photoresist pattern begins to blur at about 45 nanometer (nm) half pitch. To continue to use fabrication equipment purchased for larger technology nodes, multiple patterning methods have been developed.

Multiple patterning technology (MPT) involves forming patterns on a single layer over a substrate using two or more different masks in succession. As long as the patterns within each individual mask comply with the relevant minimum separation distances for the technology node, the combination of patterns formed using the plural masks may include smaller spacings than the minimum separation distance design rule. Thus, MPT provides flexibility and generally allows for significant reduction in overall IC layout.

MPT is a layout splitting method analogous to an M-coloring problem for layout splitting in graph theory, where M is the number of masks used to expose a single layer (and the number of exposures). For example, if two masks are to be used (double patterning technology, DPT), it is customary to refer to the patterns as being assigned one of two “color types”, where the color corresponds to a photomask assignment.

Some multi-patterning methods, such as the litho-etch-litho-etch (LELE) method use plural reticles in succession for patterning a single layer. Other multi-patterning methods, such as the self-aligned double patterning (SADP) method, use one reticle as a first mask to pattern a resist, and then form spacers adjacent those patterns, and use the spacers as a hard mask for further etching.

BRIEF DESCRIPTION OF THE DRAWINGS

Aspects of the present disclosure are best understood from the following detailed description when read with the accompanying figures. It is noted that, in accordance with the standard practice in the industry, various features are not necessarily drawn to scale. In fact, the dimensions of the various features may be arbitrarily increased or reduced for clarity of discussion.

FIG. 1A illustrates a circuit layout including a plurality of conductive lines, in accordance with some embodiments.

FIG. 1B illustrates a graph-representation of the circuit layout of FIG. 1A, in accordance with some embodiments.

FIG. 2 is a flow chart illustrating a triple patterning checking method, in accordance with some embodiments.

FIGS. 3A-3F illustrates a graph representative of a circuit layout having the triple patterning checking method of FIG. 2 applied thereto, in accordance with some embodiments.

FIG. 4 illustrates a graph representative of a circuit layout having a K₃₋₁ reduction process iteratively applied thereto, in accordance with some embodiments.

FIG. 5 illustrates a graph representative of a circuit layout having a square-loop reduction process applied thereto, in accordance with some embodiments.

FIG. 6 is a flow chart illustrating a method of forming photomasks for a triple patterning photolithographic process, in accordance with some embodiments.

FIG. 7 is a flow chart illustrating a quadruple patterning checking method, in accordance with some embodiments.

FIGS. 8A and 8B illustrate graphs representative of circuit layouts having the quadruple patterning checking method of FIG. 7 applied thereto, in accordance with some embodiments.

FIGS. 9A-9B are a flow chart illustrating a method of forming photomasks for a quadruple patterning photolithographic process, in accordance with some embodiments.

FIG. 10 is a flow chart illustrating a n-pattern checking method, in accordance with some embodiments.

FIG. 11 illustrates a system 400 for generating a plurality of photomasks according to one or more the methods disclosed herein, in accordance with some embodiments.

DETAILED DESCRIPTION

This description of the exemplary embodiments is intended to be read in connection with the accompanying drawings, which are to be considered part of the entire written description. In the description, relative terms such as “lower,” “upper,” “horizontal,” “vertical,”, “above,” “below,” “up,” “down,” “top” and “bottom” as well as derivative thereof (e.g., “horizontally,” “downwardly,” “upwardly,” etc.) should be construed to refer to the orientation as then described or as shown in the drawing under discussion. These relative terms are for convenience of description and do not require that the apparatus be constructed or operated in a particular orientation. Terms concerning attachments, coupling and the like, such as “connected” and “interconnected,” refer to a relationship wherein structures are secured or attached to one another either directly or indirectly through intervening structures, as well as both movable or rigid attachments or relationships, unless expressly described otherwise.

As used herein, the term “first mask” or “mask A” refers to a photomask (reticle) used in a photolithography process to expose a photoresist in a single layer over the substrate and the terms “second mask” or “mask B,” “third mask” or “mask C,” “fourth mask” or “mask D,” etc., can refer to additional photomasks (or hard mask) used to pattern the same layer as the first mask (referred to herein collectively as “the masks” or “plurality of masks”). The masks are used to pattern sets of polygons (circuit patterns, such as conductive lines) in the same layer over the substrate, with each mask corresponding to a separate set of polygons. For example, three separate sets of patterns such as conductive lines are denoted as A, B, and C respectively. The additional masks are used to pattern additional groups of lines (or other polygons) using a process that can be the same and/or different from the process used to pattern a first group of lines using the first mask.

As used herein, the term “color” or “graph color” is used to refer to one or more polygons (e.g., circuit patterns) selected for inclusion on one of the plurality of masks. Graph coloring is a special case of graph labeling that assigns “colors” to elements of a graph subject to constraints, for example, polygons that cannot be included on the same photomask. In some embodiments, vertices are colored. Vertex coloring, or proper vertex coloring, includes assignment of colors to vertices within a graph such that no two vertices sharing the same edge have the same color. In some embodiments, each of the polygons in a circuit layout are considered a vertex within a graph and can be assigned a color. For example, a “first color” can refer to one or more polygons selected for inclusion on a first mask, a “second color” can refer to one or more polygons selected for inclusion on a second mask, a “third color” can refer to one or more polygons selected for inclusion on a third mask, etc. In some embodiments, edges between the vertices can be colored. As used herein, the term “coloring check” can be used to refer to a process of checking compatibility of a graph with a selected coloring process and/or the process of coloring a graph according to a selected coloring scheme.

As used herein, the notation K_(n) is used to refer to one or more graphs having a predetermined number of vertices n connected by a plurality of edges to define an outer perimeter shape. For example, the notation K₃ refers to a graph having three vertices each connected by a perimeter edge to define a triangular (or three-sided) shape, the notation K₄ refers to a graph having four vertices each connected by a perimeter edge to define a four-sided shape (such as a square, kite, non-regular four-sided shape, etc.), K₅ refers to a graph having five vertices each connected by a perimeter edge to define a five-sided shape, etc. Further, as used herein, the notation K_(n−1) reduction is used to denote the reduction of a K_(n) graph to a K_((n−1)) graph by removing one of the vertices from the K_(n) graph. For example, the notation K₄₋₁ reduction refers to a reduction process of removing a first vertex from a K₄ graph (i.e., a graph having four vertices each connected by a perimeter edge to define a four-sided shape) to form a K₃ graph.

FIG. 1A illustrates one embodiment of a semiconductor circuit layout 2 including a plurality of conductive lines (or traces) 4 a-4 e. The conductive lines 4 a-4 e correspond to one or more semiconductor elements, such as, power lines, bit lines, word lines, gates, and/or any other suitable semiconductor element. For example, in the illustrated embodiment, a first conductive line 4 a, a second conductive line 4 b, and a fifth conductive line 4 e are connected to first voltage (VD) and a third conductive line 4 c and a fourth conductive line 4 d are connected to a second voltage (VG). The conductive lines 4 a-4 e are arranged in a predetermined layout with each of the conductive lines 4 a-4 e having a spacing 6 a-6 j therebetween. The spacing lines 6 a-6 j are shown in phantom as they are added purely for illustrative purposes and are not part of the circuit layout 2. The circuit layout 2 can be represented as a graph 8 as shown in FIG. 1B. The conductive lines 4 a-4 e are each represented as a vertex 10 a-10 e in the graph 8. If the distance 6 a-6 j between a first conductive line 4 a-4 e and a second conductive line 4 a-4 e is less than a predetermined distance, an edge 12 a-12 j is added between the vertices 10 a-10 e in the graph 8.

For example, as shown in FIG. 1A, spacing 6 a, 6 e-6 g between the first conductive line 4 a and each of a second conductive line 4 b, a third conductive line 4 c, a fourth conductive line 4 d, and a fifth conductive line 4 e is less than a predetermined minimum spacing. Edges 12 a, 12 e-12 g are added to the graph 8 to connect the first vertex 10 a and each of the second vertex 10 b, the third vertex 10 c, the fourth vertex 10 d, and the fifth vertex 10 e respectively. Similarly, spacing 6 a-6 b, 6 h-6 i between the second conductive line 4 b and each of the first conductive line 4 a, the third conductive line 4 c, the fourth conductive line 4 d, and the fifth conductive line 4 e are less than predetermined minimum spacing. Edges 12 a-12 b, 12 h-12 i are added to the graph 8 to connect the second vertex 10 b to each the first vertex 10 a, the third vertex 10 c, the fourth vertex 10 d, and the fifth vertex 10 e. The third vertex 10 c, the fourth vertex 10 d, and the fifth vertex 10 e each include edges 12 a-12 j connecting the vertices 10 c-10 e to each of the other vertices 10 a-10 e, representing a spacing less than the predetermined minimum spacing between each of the conductive lines 4 a-4 e in the circuit layout 2. In some embodiments, a length of the edge 12 a-12 j corresponds to the distance between the connected vertices 10 a-10 e.

In some embodiments, each of the vertices 10 a-10 e are assigned a “color” corresponding to the inclusion of the associated conductive line 4 a-4 e on a photomask during a photolithographic process of forming the circuit layout 2 on a semiconductor substrate. For example, in embodiments corresponding to a three-photomask process, each of the vertices 10 a-10 e are assigned one of three selected colors, while in a four-photomask process, each of the vertices 10 a-10 e are assigned one of four selected colors. FIG. 1B illustrates the graph 8 having a four-color coloring. Although the term “color” is used herein, it will be appreciated that each of the vertices 10 a-10 e (and corresponding conductive lines 4 a-4 e) can be assigned using any suitable categories indicative of inclusion on a photomask and are not limited to the use of colors. For example, in various embodiments, the “colors” can include letter groups, number groups, word groups, color groups, and/or any other suitable grouping corresponding to the number of photomasks used during a photolithographic process. In FIG. 1B, the colors are represented by various line patterns. For example, the first vertex 10 a is assigned a first color, the second vertex 10 b and the fifth vertex 10 e are both assigned a second color, the third vertex 10 c is assigned a third color, and the fourth vertex 10 d is assigned a fourth color.

In order to convert the circuit layout 2 into a plurality of photomasks corresponding to a photolithographic process, each of the vertices 10 a-10 e must be assigned a color such that no two connected vertices 10 a-10 e (i.e., no two vertices 10 a-10 e having an edge 12 therebetween) share the same color. A color selected from a predetermined number N of colors is assigned to each of the vertices 10 a-10 e. The predetermined number N is equal to the number of photomasks used in a photolithographic process and each of the colors corresponds to one of the photomasks. For example, if a triple patterning photolithographic process is used, the predetermined number of colors is 3 (N=3).

If each of the vertices 10 a-10 e can be assigned one of the predetermined M colors such that no two connected vertices 10 a-10 e share a color, the circuit layout 2 is compatible with a photolithographic process using M or fewer photomasks. If two vertices 10 a-10 e are connected by an edge and share a color, a conflict is exists within the graph 8 and the circuit layout 2 is not compatible with a photolithographic process using M or fewer photomasks. For example, as shown in FIG. 1B, the graph 8 representative of the circuit layout 2 has an edge 12 h connecting a second vertex 10 b and a fifth vertex 10 e that have the same color. The graph 8 is not four-color colorable, and therefore is not compatible with photolithographic processes using four or fewer photomasks. Circuit layouts 2 having conflicts must be redesigned to remove the conflicts and/or be assigned to a photolithographic process having a greater number of photomasks.

In some embodiments, a graph 8 is checked for compatibility with a N-photomask photolithographic process prior to producing photomasks for the photolithographic process. The coloring check can be performed using any suitable coloring process, such as, for example, a brute-force coloring process, a heuristic coloring process, a deterministic rules-based process, and/or any other suitable coloring process utilized in graph theory. In some embodiments, a multi-patterning checking flow can be applied prior to performing the coloring process to reduce the complexity of color patterning for multi-patterning (i.e., multiple photomask) circuit layouts.

FIG. 2 illustrates a triple patterning checking flow method 100 a, in accordance with some embodiments. The triple patterning checking flow method 100 a determines whether a circuit layout is compatible with a three-photomask photolithographic process. At step 102, a circuit layout is generated using one or more known circuit layout techniques. The circuit layout can be represented as a graph having a plurality of vertices (representing each of the conductive lines) and a plurality of edges (representing spacing between the conductive lines that is less than predetermined minimum spacing). FIGS. 3A-5 illustrate various embodiments of graphs 50, 80 a, 90 a representative of one or more circuit layouts. For example, as shown in FIG. 3A, graph 50 includes a plurality of vertices 52 a-52 o and a plurality of edges 54 a-54 z representative of a first circuit layout (not shown). As another example, as shown in FIG. 4, graph 80 a includes a plurality of vertices 82 a-82 i and a plurality of edges 84 a-84 r representative of a second circuit layout (not shown). At step 104, any vertices having less than three edges (i.e., two or fewer edges) are removed from the graph. Vertices having less than three edges can be removed, as such vertices can definitively be colored with one of the three colors without being adjacent to a vertex having the same color. For example, as shown in FIG. 3A, a first vertex 52 a has a first edge 54 a connecting to a second vertex 52 b and a second edge 54 b connecting to a sixth vertex 52 f. The first vertex 52 a can be assigned one of three colors color based on the colors assigned to the second vertex 52 b and the sixth vertex 52 f without potentially generating any conflicts in the graph 50. Because the color of the first vertex 52 a can be determined solely from the colors of the second and sixth vertices 52 b, 52 f, the first vertex 52 a can be ignored for the purposes of the triple patterning checking flow method 100 a. FIG. 3B illustrates a reduced graph 50 a after removal of the first vertex 52 a.

At step 106, the graph is partitioned. Partitioning a graph includes separating the graph at single vertex connections to generate two or more separate, partial graphs. For example, as shown in FIG. 3B, the graph 50 a can be separated at an third edge 54 c extending between a sixth vertex 52 c and a thirteenth vertex 52 m to generate a first partial graph 60 a and a second partial graph 70 a, as shown in FIG. 3C. Graph partitioning is described in greater detail in U.S. Pat. No. 9,122,838, issued on Sep. 1, 2015, and entitled “Triple-Pattern Lithography Layout Decomposition,” the disclosure of which is incorporated herein by reference in its entirety.

At step 108, a reduction process is applied to each of the partial graphs 60 a, 70 a to further reduce the complexity of the graph 60 a, 70 a prior to performing a triple patterning conflict check (e.g., three color coloring). The reduction process 108 reduces processing time and/or prevents overkill (e.g., over application of rules/brute force processing) for a triple patterning checking process. The reduction process 108 can include a K₄₋₁ reduction step 110 and a square loop reduction step 112. The K₄₋₁ reduction step 110 reduces one or more kites (K₄₋₁) in the graph to a triangle by combining two vertices in the kite. The partial graph 70 a, illustrated in FIG. 3C, is one example of a kite (K₄). The kite (K₄) 70 a includes four vertices 52 l-52 o and a plurality of edges 54 u-54 z. A first set of the plurality of edges 54 v-54 x connect the vertices 52 m-52 o in series to define a kite or square shape. A second set of the plurality of edges 54 u, 54 y extends between non-adjacent vertices.

A K₄₋₁ reduction reduces a kite (K₄) to a triangle (K₃) by merging a first vertex into a second, non-adjacent vertex. For example, as shown in FIG. 3C, the twelfth vertex 52 l is located non-adjacent to the fourteenth vertex 52 n. That is, the twelfth vertex 52 l and the fourteenth vertex 52 n are not connected by one of the first set of edges 54 v-54 x, 54 z defining the perimeter of the kite. In some embodiments, the selected vertices are vertices not connected by an edge. That is, the selected vertices can be assigned the same color during triple patterning. The edge connections of the removed vertex 52 n are added to the vertex 52 l into which the removed vertex 52 n is merged. For example, as shown in FIG. 3C, the fourteenth vertex 52 n has edge connections 54 u, 54 w, 54 z to each of the other vertices 52 l-52 m, 52 o in the kite graph 70 a. The edge connections of the fourteenth vertex 52 n are added to the twelfth vertex 52 l when the fourteenth vertex 52 n is removed from the graph 70 a. If the remaining vertex 52 l contains connections to the same vertices as the merged vertex 52 n, the edges are not added to the remaining vertex 52 l. For example, the twelfth vertex 52 l has edge connections 54 r, 54 x to the thirteenth vertex 52 m and the fifteenth vertex 52 o prior to the merger. Therefore, the edges 54 w, 54 z between the fourteenth vertex 52 n and each of the thirteenth vertex 52 m and the fifteenth vertex 52 o are not added to the twelfth vertex 52 l, as they are subsumed within the already existing edges 54 u, 54 x. If the merged vertices 52 l, 52 n are connected by an edge 54 u, the edge 54 u becomes a loop 54 u′ connecting to the remaining vertex 52 l, as shown in FIG. 3F. If a loop 54 u′ is generated during a K₄₋₁ reduction, the graph 70 a cannot be colored using three colors and the therefore is not triple patterning compliant. For example, the reduced graph 70 b includes a loop edge 54 u, indicating that the original partial graph 70 a is not three-color colorable. The circuit layout corresponding to the partial graph 70 a is not compatible with a triple patterning lithographic process due to spacing between the conductive lines, and at least the partial graph 70 a of the original graph 50 must be redesigned. In some embodiments, a K₄₋₁ reduction step 110 can be iteratively executed to reduce additional K₄ graphs. For example, FIG. 3D is a reduced graph 60 b illustrating the partial graph 60 a after undergoing two K₄₋₁ reductions (as discussed in greater detail below with respect to FIG. 6).

In some embodiments, the reduction process at step 108 includes a square loop reduction step 112. As used herein, square loops are defined by four vertices connected by a plurality of edges to define a perimeter without any additional edges between non-adjacent vertices and do not necessarily require the perimeter to define a square shape. FIG. 5 illustrates a graph 90 a including two square loops 96 a, 96 b, in accordance with some embodiments. The graph 90 a includes a first square loop 96 a defined by a first vertex 92 a, a third vertex 92 c, a fifth vertex 92 e, and a sixth vertex 92 f. A first edge 94 a connects the first vertex 92 a to the third vertex 92 c, a second edge 94 b connects the third vertex 92 c to the fifth vertex 92 e, a third edge 94 c connects the fifth vertex 92 e to the sixth vertex 92 f, and a fourth edge 94 d connects the sixth vertex 92 f to the first vertex 92 a. The graph 90 a further includes a second square loop 96 b defined by the second vertex 92 b, the third vertex 92 c, the fourth vertex 92 d, and the fifth vertex 92 e. The second edge 94 b connects the third vertex 92 c to the fifth vertex 92 e, a fifth edge 94 e connects the fifth vertex 92 e to the fourth vertex 92 d, a sixth edge 94 f connects the fourth vertex 92 d to the second vertex 92 b, and a seventh edge 94 g connects the second vertex 92 b to the third vertex 92 c.

A square loop 96 a, 96 b can be reduced by selecting one pair of diagonal nodes for combination. The selected pair of diagonal vertices can be the diagonal vertices having the largest separation as defined by the sum of the two edges separating the vertices (e.g., the length of each edge connecting the diagonal vertices in series is greater than the length of each of the edges connecting other diagonal vertices in series). For example, reduced graph 90 b illustrates the graph 90 a after the first square loop 96 a is reduced by merging the first vertex 92 a and the fifth vertex 92 e. The edge connections of the merged vertices 92 e are maintained. For example, the fifth vertex 92 e is connected to the third vertex 92 c, the fourth vertex 92 d, and the sixth vertex 92 f prior to being merged with the first vertex 92 a. The first vertex 92 a has edge connections to the second vertex 92 b, the third vertex 92 c, and the sixth vertex 92 f prior to the merger. After merging the vertices 92 a and 92 e, the remaining vertex, the first vertex 92 a, has edge connections 94 a, 94 d, 94 i, 94 e′ to the second vertex 92 b, the third vertex 92 c, the fourth vertex 92 d and the sixth vertex 92 f. The edge 94 e′ between the first vertex 92 a and the fourth vertex 92 d is added to the first vertex 92 a to reflect the original edge connection 92 e between the fourth vertex 92 d and the merged fifth vertex 92 e.

As another example, reduced graph 90 c illustrates the graph 90 a after the second square loop 96 b is reduced by merging the fourth vertex 92 d into the third vertex 92 c. The fourth vertex 92 d has edge connections to the second vertex 92 b, the fifth vertex 92 e, and the sixth vertex 92 f prior to being merged with the second vertex 92 b. The third vertex 92 c has edge connections to the first vertex 92 a, the second vertex 92 b, and the fifth vertex 92 e. After merging the vertices 92 c and 92 d, the remaining vertex, the third vertex 92 c, has edge connections 94 a, 94 b, 94 g, 94 h′ to the first vertex 92 a, the second vertex 92 b, the fifth vertex 92 e, and the sixth vertex 92 f The edge 94 h′ between the second vertex 92 b and the sixth vertex 92 f is added to the second vertex 92 a to reflect the original edge connection 94 h between the sixth vertex 92 f and the merged fourth vertex 92 d.

After reducing the graph at step 108, a triple patterning check is performed on the reduced graph at step 114. The triple patterning check can include any suitable triple patterning check, such as, for example, a brute-force triple coloring process, a heuristic triple coloring process, a rules-based triple coloring process, and/or any other suitable triple coloring process. If the reduced graph, such as reduced graph 60 b, 60 c, 90 b, 90 c passes the triple patterning check, the original graph is compatible with a triple patterning lithography process.

In some embodiments, the triple patterning checking method 100 a is iterated until the graph is reduced as far as possible. For example, as shown in FIG. 4, an original graph 80 a can be reduced to the reduced graph 80 d by two iterations of a K₄₋₁ reduction process. If the reduction process of step 108 is further applied to the reduced graph 80 d, the graph can be reduced to nothing as described in greater detail below, indicating that the original graph 80 a is compatible with a three-color coloring. In some embodiments, the method 100 a is iterated until original graph 80 a is reduced to nothing (indicating a passed result) or until the original graph 80 a cannot be further reduced, at which point the triple patterning check at step 114 is performed.

FIG. 4 illustrates one embodiment of the triple patterning checking method 100 a as applied to a graph 80 a. The graph 80 a includes a plurality of vertices (or nodes) 82 a-82 j representative of a plurality of conductive traces and a plurality of edges 84 a-84 r representing spacing less than a minimum predetermined spacing between the conductive lines. The triple patterning checking method 100 a is applied to the graph 80 a. There are no nodes with less than three edge connections, so the method 100 a proceeds to reduction process 108. At step 110, one or more kites (K₄) are reduced. For example, in some embodiments, two kites 86 a, 86 b are reduced by a K₄₋₁ reduction process at step 110. The first kite 86 a includes a first vertex 82 a, a second vertex 82 b, a fourth vertex 82 d, and a fifth vertex 82 e. The perimeter of the first kite 86 a is defined by a plurality of edge connections 84 a, 84 c, 84 e, 84 h. The second vertex 82 b and the fourth vertex 82 d are coupled by an internal edge 84 d. Similarly, the second kite 86 b includes a seventh vertex 82 g, an eighth vertex 82 h, a ninth vertex 82 i, and a tenth vertex 82 j. The perimeter of the second kite 86 b is defined by a plurality of edge connections 84 m, 84 o, 84 p, 84 r and the kite 86 b includes an edge connection 84 q between the seventh vertex 82 g and the tenth vertex 82 j.

The identified kites 86 a, 86 b are reduced by a K₄₋₁ reduction process. With respect to the first kite 86 a, the fifth vertex 82 e is merged into the first vertex 82 a. Similarly, with respect to the second kite 86 b, the eighth vertex 82 h is merged with the ninth vertex 82 i. In the illustrated embodiment, the vertices 82 a, 82 e, 82 h, 82 i selected for combination are non-adjacent vertices not having an edge connection therebetween, e.g. are capable of being colored the same color. The edge connections of the merged vertices 82 e, 82 h are added to the remaining vertices 82 a, 82 i. An additional edge 84 s is added between the first vertex 82 a and the ninth vertex 82 i to illustrate the edge connection 84 k between the removed vertices 82 e, 82 h.

In some embodiments, after removing the identified kites 86 a, 86 b the first reduced graph 80 b can be further reduced according to one or more steps of the method 100 a. For example, the first reduced graph 80 b includes a second vertex 82 b and a tenth vertex 82 j which each have two edge connections after the K₄₋₁ reduction. The second vertex 82 b and the tenth vertex 82 j can be removed from the first reduced graph 80 b according to step 104 to generate a second reduced graph 80 c. Additional reduction, such as one or more additional K₄₋₁ reductions can be performed to further reduce the graph 80 c. For example, in some embodiments, an additional K₄₋₁ reduction can be applied to the second reduced graph 80 c to remove a third kite 86 c and a fourth kite 86 d generated during the prior reductions. The additional K₄₋₁ reduction generates a reduced triangular graph 80 d.

The reduced triangular graph 80 d includes only vertices 82 a, 82 c, 82 d having two edges, and therefore the reduced triangular graph 80 d can be reduced to an empty graph according to step 104. The original graph 80 a passes the triple patterning check at step 114 and the circuit layout represented by the original graph 80 a is compatible with a triple-photomask photolithographic process. Each of the remaining vertices 82 a, 82 c, 82 d in the reduced graph 80 d are assigned one of three colors according to a triple patterning coloring process. Each of the merged vertices 82 a-82 j of the original graph 80 a are unpacked from the reduced graph 80 d while maintaining the same color as a corresponding vertex 82 a, 82 c, 82 d in the reduced graph 80 d. For example, when the first kite 86 a was reduced, the fifth vertex 82 e was merged into the first vertex 82 a. If the first vertex 82 a is assigned a first color, the fifth vertex 82 e is also assigned the first color and can be unmerged (i.e., added back to the graph 80 d). In some embodiments, colors are assigned to merged vertices without adding the merged vertices back to the graph 80 d. As another example, during reduction of the second kite 86 b, the eighth vertex 82 h is merged into ninth vertex 82 i. The ninth vertex 82 i can subsequently merged into the fourth vertex 82 d during a later K₄₋₁ reduction. The fourth vertex 82 d can be assigned a third color. The ninth vertex 82 i, which was merged into the fourth vertex 82 d and the eighth vertex 82 h, which was merged into the ninth vertex 82 i, are each also assigned the third color. Each of the conductive lines in the circuit layout corresponding to the graph 80 a can be assigned to one of three photomasks based on the color assigned to the respective vertex 82 a-82 j.

FIG. 6 illustrates one embodiment of a method 100 b of generating a first photomask, a second photomask, and a third photomask for use in a triple patterning lithography process, according to some embodiments. The method 100 b includes steps similar to those discussed above with respect to method 100 a, and similar description is not repeated herein. The method 100 b is discussed herein with respect to FIGS. 3A-3F and 6. A circuit layout including a plurality of conductive lines is generated at step 116. At step 102, the circuit layout is converted to a graph, such as graph 50, including a plurality of vertices 52 a-52 o representing each of the conductive lines in the circuit layout. A plurality of edges 54 a-54 z are indicative of a spacing less than a predetermined minimum spacing between the conductive lines in the circuit layout.

At step 104, each vertex 54 a having less than three edges is removed from the graph 50 to generate a first reduced graph 50 a. As noted above, vertices having two or fewer edges can be definitively colored with one of the three colors without being adjacent to a vertex having the same color. At step 118, a check is performed to determine if any of the remaining vertices 52 b-52 o have two or fewer edges as a result of the removal of the vertex 52 a. If a vertex with two or fewer edges is identified, the method 100 b returns to step 104. If none of the remaining vertices 52 b-52 o have two or fewer edges, the reduced graph 50 a is partitioned at step 106. The partitioning step 106 separates the graph 50 a into a first partial graph 60 a and a second partial graph 70 a. Each of the partial graphs 60 a, 70 a are separately checked by the method 100 b. At step 120, the method 100 b determines whether the partitioning of step 106 has generated any additional connected components. If edges are removed from one or more components during partitioning, the method 100 b returns to step 104. If the graph cannot be further partitioned and/or edges are not removed, the method 100 b proceeds to a K₄ reduction process 108.

A K₄ reduction process is performed to reduce each of the partial graphs 60 a, 70 a to one or more simplified graphs. Simplifying the partial graphs 60 a, 70 a reduces processing time and/or overkill for a subsequent triple patterning checking. At step 122, the reduction process determines whether any kites (K₄) exist within the partial graphs 60 a, 70 a. For example, the first partial graph 60 a includes at least two kites 56 a, 56 b. The first kite 56 a is defined by the second vertex 52 b, the third vertex 52 c, the fourth vertex 52 d, and the fifth vertex 52 e. The second kite 56 b is defined by the seventh vertex 52 g, the eighth vertex 52 h, the tenth vertex 52 j, and the eleventh vertex 52 k. As another example, the second partial graph 70 a is a kite graph 56 c defined by the twelfth vertex 52 l, the thirteenth vertex 52 m, the fourteenth vertex 52 n, and the fifteenth vertex 52 o.

If a kite 56 a-56 c (K₄) is identified in one of the partial graphs 60 a, a K₄₋₁ reduction is performed at step 110. For example, the first partial graph 60 a is reduced to by performing a K₄₋₁ reduction on each of the first and second kites 56 a, 56 b. The first kite 56 a is reduced by merging the fourth vertex 52 d into the second vertex 52 b and the second kite 56 b is reduced by merging the seventh vertex 52 f into the eleventh vertex 52 k. After merging the vertices 52 b, 52 d, 52 f, 52 k of the first kite 56 a and the second kite 56 b respectively, the partial graph 60 a would include vertices 52 c, 52 j having two or fewer edge connections (not shown). These vertices 52 c, 52 j can be removed according to step 104 to generate the reduced graph 60 b of FIG. 3D. The edge connections of each of the merged vertices 52 d, 52 f are added to the remaining vertices 52 b, 52 k. Similarly, the second partial graph 70 a is reduced by merging the fourteenth vertex 52 n into the twelfth vertex 52 l to generate the reduced graph 70 b of FIG. 3F. The edge connections of each of the merged vertex 52 l are added to the remaining vertex 52 o. A new edge 58 is added to the partial graph 60 b between the remaining vertices 52 b, 52 k to represent the edge connection 52 k between the removed vertices 52 d, 52 g.

At step 124 the method 100 b determines whether any edges (or links) connect to a single node (i.e., define a loop) or are empty at one or more sides (i.e., do not connect to a vertex at one or more sides). For example, as shown in FIG. 3F, the reduced graph 70 b includes an edge 54 u′ defining a loop at the twelfth node 52 l. If the method 100 b determines that there are loop edges 54 u′, the method 100 b generates a failure for at least the partial graph 70 a at step 132. A failure indicates that the proposed circuit layout represented by at least the partial graph 70 a is not compatible with triple patterning lithography and must be redesigned and/or assigned to a higher-number patterning process. In some embodiments, the method 100 b processes the first partial graph 60 a and the second partial graph 70 a separately, such that a failure of one partial graph 70 a does not preclude passing another partial graph 60 a. In other embodiments, failure of a first partial graph 70 a causes failure of the entire original graph 50.

If the method 100 b determines that at least one reduced graph 60 b does not include single node or empty node edges at step 124, the method 100 b can perform an optional square loop reduction process 126. In some embodiments, the method 100 b skips the square loop reduction process 126 and proceeds directly to step 130. If a square loop reduction process 126 is performed, a square loop check is performed at step 128 to identify one or more square loops in a graph. If one or more square loops are identified, a square loop reduction can be performed at step 112. For example, FIG. 5 is a graph 90 a including at least two square loops 96 a, 96 b, in accordance with some embodiments. The first square loop 96 a is defined by a first vertex 92 a, a third vertex 92 c, a fifth vertex 92 e, and a sixth vertex 92 f and the second square loop 96 b is defined by the second vertex 92 b, the third vertex 92 c, the fourth vertex 92 d, and the fifth vertex 92 e. One or more of the square loops 96 a, 96 b can be reduced by merging non-adjacent vertices having a longest path distance therebetween, according to step 112 (as described above). For example, a first square-loop reduced graph 90 b can be generated by merging the fifth vertex 92 e into the first vertex 92 a. As another example, a second square-loop reduced graph 90 c can be generated by merging the fourth vertex 92 d into the third vertex 92 c. Although examples are shown merging only one of the square loops 96 a, 96 b, it will be appreciated that both square loops 96 a, 96 b can be reduced during a square loop reduction process 126.

Referring back to FIGS. 3A-3F and 6, in some embodiments after performing the square loop reduction process 126, the method 100 b returns to step 104 and iterates the triple-patterning reduction process 100 b on a reduced graph 60 b and/or a square loop reduced graph 60 c. In some embodiments, the method 100 b is performed iteratively until either the original graph 50 is reduced to nothing (indicating a positive checking result) or cannot be further reduced. If the original graph 50 cannot be further reduced, the method 100 b generates a final simplified graph at step 130 and performs a triple patterning conflict check on the simplified graph at step 114. In some embodiments, the method 100 b proceeds to step 130 after a predetermined number of iterations through the reduction process 108.

The triple patterning check at step 114 can be any suitable triple patterning check, such as, for example, a brute-force triple patterning check, a heuristic triple patterning check, a DRC deterministic check, and/or any other suitable check. A check result is proved at step 132 based on the triple patterning check. If the check result is positive, a plurality of photomasks are generated for a photolithographic process at step 134. In some embodiments, three photomasks are generated for a triple patterning photolithographic process. Each of the photomasks include the conductive lines assigned the same color. For example, in some embodiments, the first photomask includes conductive lines having a corresponding vertex assigned a first color, the second photomask includes conductive lines having a corresponding vertex assigned a second color, and the third photomask includes conductive lines having a corresponding vertex assigned a third color. The photomasks can be formed using one or more known processes. If the check result is negative, the circuit layout is identified for revision.

FIG. 7 is a flow chart illustrating a quadruple patterning checking method 200 a, in accordance with some embodiments. The quadruple patterning checking method 200 a is similar to the triple patterning checking flow method 100 a described above and similar description is not repeated herein. The quadruple patterning checking flow method 200 a is configured to check a circuit layout for compatibility with a quadruple patterning lithography process that utilizes four photomasks for patterning a circuit. FIGS. 8A and 8B illustrate various graphs 250 a, 260 a having a quadruple checking method 200 a applied thereto.

At step 204, vertices having fewer than 4 (i.e., three or fewer) edge connections are removed from the graph 250 a, 260 a. Step 204 is similar to step 104 of method 100 a, but expanded to allow for the use of four colors in a quadruple patterning check. For example, as shown in FIG. 8A, the graph 250 a includes a first vertex 252 a and a third vertex 252 c each having only three edges 254 a, 254 e, 254 h and 254 b, 254 c, 254 i extending respectively therefrom. Each of the vertices 252 a, 252 c can be removed from the graph, as the vertices 252 a, 252 c can definitively be colored with one of the four colors without being adjacent to a vertex having the same color. A reduced graph 250 b can be generated by removing the first vertex 252 a and the third vertex 252 c. The reduced graph 250 b is a triangle graph including vertices 252 b, 252 d, and 252 e having only three or fewer edge connections and can be further reduced to an empty graph according to step 204, indicating that the graph 250 a passes the quadruple patterning check and is compatible with a quadruple photomask lithographic process (as discussed in more detail below).

If the method 200 a is performing a first iteration, an initial process 206 is performed. On subsequent iterations through the method 200 a, the initial process 206 is skipped. During the initial process, each component in the graph (after removal of all vertices having less than four edges) are recorded at step 208. A check is performed at step 210 to determine whether any K₄ graphs exist, such as kite graphs and/or square loop graphs. If no K₄ graphs are identified, the graph can be reduced using a triple patterning reduction. The method 200 a proceeds to a triple patterning reduction process 108. The triple patterning reduction process 108 is described above in conjunction with FIGS. 1-6, and similar description is not repeated herein. The triple patterning reduction process 108 reduces K₄ graphs in the original graph 250 a, 260 a. In some embodiments, the triple patterning reduction process 108 can completely reduce an original graph, such as original graph 250 a, indicating that the original graph 250 a is compatible with a triple lithographic process and does not require use of a quadruple pattern lithographic process. In other embodiments, the triple patterning reduction process 108 partially reduces the original graph 250 a, 260 a and further processing is performed by the method 200 a to verify compatibility with the quadruple patterning process.

After performing the triple patterning reduction process 108, the method 200 a checks the graph 260 a at step 216 to determine whether the graph 260 a is empty. If the graph is empty, the method proceeds to step 214 to perform a quadruple coloring check, as described in more detail below. As noted above, a coloring check, such as a quadruple coloring check, can include a process of coloring one or more vertices of a graph. In embodiments in which the graph is reduced to an empty graph, the quadruple coloring check performs a quadruple coloring on the partial graph that is generated just prior to the empty graph. For example, as shown in FIG. 8A, the partial graph 250 b includes three vertices 252 b, 252 d, 252 e each only a single edge connection 254 d, 254 f, 254 g. The partial graph 250 b can be reduced to an empty graph according to step 206. If the graph 250 b is reduced to an empty graph, the method 200 a proceeds to step 214 and performs a quadruple coloring of the partial graph 250 b, which was the last graph generated prior to the empty graph. If the graph 260 a is not empty and/or the method 200 a determines that the graph 260 a contains one or more K₄ graphs at step 210, the method 200 a proceeds to step 212.

At least one K₅₋₁ reduction is performed at step 212. A K₅ graph includes five vertices 262 a-262 e each having an edge connection 264 a-264 e defining an outer perimeter and at least one inner edge connection 264 f-264 i. For example, as shown in FIG. 8B, the graph 260 a contains at least one K₅ graph 266. The K₅ graph 266 includes a first vertex 262 a, a second vertex 262 b, a third vertex 262 c, a fourth vertex 262 d, and a fifth vertex 262 e. The first vertex 262 a is connected to the second vertex 262 b by a first edge 264 a, the second vertex 262 b is connected to the third vertex 262 c by a second edge 264 b, the third vertex 262 c is connected to the fourth vertex 262 d by a third edge 264 c, the fourth vertex 262 d is connected to the fifth vertex 262 e by a fourth edge 264 d, and the fifth vertex 262 e is connected to the first vertex 262 a by a fifth edge 264 e. A sixth edge 264 f connects the first vertex 262 a and the fourth vertex 262 d, a seventh edge 264 g connects the first vertex 262 a and the third vertex 262 c, an eighth edge 264 h connects the fifth vertex 262 e and the second vertex 262 b, and a ninth edge 264 i connects the fourth vertex 262 d and the second vertex 262 b.

A K₅₋₁ reduction reduces the K₅ graph 266 by combining a first of the vertices 262 a-262 e of the K₅ graph 266 with a second of the vertices 262 a-262 e of the K₅ graph 266. For example, in some embodiments, the fifth vertex 262 e is merged into the third vertex 262 c. In some embodiments, the selected vertices 262 c, 262 e are not directly connected by an edge and can be assigned the same color during a quadruple coloring process. As shown in FIG. 8B, the graph 260 a is reduced to a simplified graph 260 b having only five vertices 262 a-262 d, 262 f After performing the K₅₋₁ reduction, the method 200 a can return to step 206 and remove any vertices having three or fewer edge connections. If four or fewer vertices remain in the graph after removing vertices at step 206, the graph 260 a is compatible with a quadruple photomask lithographic process and the method 200 a can proceed to step 214 to perform a quadruple coloring of the remaining vertices and/or the original graph 260 a.

In some embodiments, if the graph 260 b cannot be further reduced, either by a K₅₋₁ reduction, a K₄₋₁ reduction, and/or removal of vertices having fewer than four edges, the method 200 a proceeds to step 214. At step 214, a quadruple patterning conflict check is performed on the reduced graph 260 b. A quadruple patterning conflict check can include any suitable check, such as a brute-force four-color coloring, a heuristic quadruple patterning conflict check, a deterministic rules-based quadruple patterning conflict check, and/or any other suitable quadruple patterning conflict check. If the reduced graph 260 b passes the quadruple patterning conflict check (i.e., the reduced graph 260 b is four-color colorable without conflicts), the associated circuit layout represented by the original graph 260 a is compatible with a quadruple photomask lithographic process. Four photomasks, each corresponding to one of the four colors used in the quadruple patterning conflict check, can be generated using one or more known methods. The four photomasks are used during a quadruple photolithographic process to form the circuit layout represented by the original graph 260 a on a semiconductor substrate.

FIGS. 9A-9B are a flow chart illustrating a method 200 b of generating photomasks for a quadruple photomask lithographic process. Circles 1, 2, and 3 illustrate connections between the steps shown in FIG. 9A and the steps shown in FIG. 9B. The method 200 b is similar to quadruple patterning checking method 200 a and the triple patterning checking flow methods 100 a, 100 b described above and similar description is not repeated herein. The method 200 b is described herein with reference to FIGS. 8B and 9A-9B. At step 116, a circuit layout is generated and/or received from a remote source. The circuit layout includes a plurality of conductive lines. The circuit layout is converted into a graph 260 a at step 202. The graph 260 a includes a plurality of vertices 262 a-262 f representative of the plurality of conductive lines. A plurality of edges 264 a-264 m representing spacing less than a minimum predetermined spacing between the conductive lines.

At step 204, vertices 262 a-262 f having three or fewer edge connections are removed from the graph 260 a. The graph 260 a does not contain any vertices having three or fewer edges, and therefore no vertices are removed at step 204. At step 218, a check is performed to determine if the removal of vertices 262 a-262 f (if any) in step 204 causes one or more additional vertices 262 a-262 f to have only three or fewer edge connections. If vertices 262 a-262 f with three or fewer edge connections are identified, the method 200 b returns to step 204. Otherwise, the graph 260 a is partitioned at step 106, if possible.

After partitioning, the graph 260 a is checked at step 212 to determine whether one or more additional connected components are generated during the partition. If additional connected components are generated, the method 200 b returns to step 204. If no additional connected components are generated, the method 200 b performs a check, at step 220, to determine if the current iteration of the quadruple patterning checking flow method 200 b is a first iteration or a subsequent iteration. If the check determines that the current iteration is a first iteration, at step 208, each of the vertices 262 a-262 f in the graph 260 a are recorded. After recording the graph 260 a, an optional second check is performed at step 224 to determine if the current iteration of the quadruple patterning checking flow method 200 b is a first iteration or a subsequent iteration. If the check determines that the current iteration is a first iteration, the method proceeds to step 210. If the check determines that the current iteration is not the first iteration, the method proceeds to step 228 (which is discussed in greater detail below).

At step 220, the method 200 b checks whether any vertices 262 a-262 f have four (K₄) or more (K₅₊) edge connections. If none of the vertices 252 a-252 f have four or more edge connections, the circuit layout corresponding to the graph 260 a can be generated by a triple patterning lithography process and therefore a quadruple patterning checking flow is not required. The method 200 b can perform a triple-pattern reduction process 108, as described above. At step 216, the method 200 b determines whether the triple patterning reduction process 108 reduced the graph 260 a to an empty graph. If so, the method 200 b outputs a positive check result at step 232, indicating that the circuit layout represented by the graph 260 a can be generated using one of a triple patterning or a quadruple patterning lithography process, depending on how the method 200 b flow reached check result step 232. If the triple patterning reduction process 108 does not reduce the graph 260 a to an empty graph, the method 200 b returns to step 208 and iterates through the method 200 b. If it is determined at step 210 that one or more vertices 262 a-262 f have four or more edge connections, the method 200 b proceeds to step 228 (which is discussed in more detail below).

If, at step 220, the check determines that the current iteration is a second or later iteration (i.e., not a first iteration), the method 200 b performs a check, at step 226 to determine whether a triple patterning reduction process 108 is currently being executed. For example, if a first iteration through the method 200 b determines that the circuit layout corresponding to the graph 260 a can be generated by a triple patterning lithography process, i.e., the method proceeded from step 210 to triple patterning reduction process 108, the check at step 226 will determine that a triple patterning reduction process 108 is currently being executed. If a triple patterning reduction process 108 is being executed, the method 200 b returns to the triple patterning reduction process 108. If the check determines that a triple patterning reduction process 108 is not being performed, the method 200 b proceeds to step 228.

At step 228, a check is performed to determine whether the graph 260 contains any K₅ graphs. For example, as shown in FIG. 8, the graph 260 a contains at least one K₅ graph 266. If a K₅ graph 266 is identified, a K₅₋₁ reduction is performed at step 212. The K₅₋₁ reduction reduces the K₅ graph 266 by merging a first of the vertices 262 e of the K₅ graph 266 with a second of the vertices 262 c of the K₅ graph 266. In some embodiments, the selected vertices 262 c, 262 e are not directly connected by an edge and can be assigned the same color during a four-color coloring check. The graph 260 a is reduced to a simplified graph 260 b. After performing the K₅₋₁ reduction, the method 200 b returns to step 204 and iterates through the method 200 b.

If the check at step 228 fails to identify any K₅ graphs, a check is performed at step 230 to determine if any of the remaining edges connect to a single node (or no nodes), i.e., are looped and/or empty. If the check identifies a looped and/or empty edge, a check result indicating a failed check and requirement to redesign the circuit is output at step 232. If no looped or empty edges are identified, a final simplified graph 260 b is generated at step 136 and provided for a quadruple patterning conflict check at step 214. The quadruple patterning conflict check can be any suitable conflict check, such as, for example, a brute-force four-color coloring, a heuristic quadruple patterning conflict check, a deterministic rules-based quadruple patterning conflict check, and/or any other suitable quadruple patterning conflict check. The output of the quadruple patterning check is provided at step 232.

If the quadruple patterning check output at step 232 is a positive result, the method proceeds to step 234 to generate a plurality of photomasks for a photolithographic process. In some embodiments, four photomasks are generated for a quadruple patterning photolithographic process. Each of the photomasks include the conductive lines assigned the same color. For example, in some embodiments, the first photomask includes conductive lines having a corresponding vertex assigned a first color, the second photomask includes conductive lines having a corresponding vertex assigned a second color, the third photomask includes conductive lines having a corresponding vertex assigned a third color, and the fourth photomask includes conductive lines having a corresponding vertex assigned a fourth color. The photomasks can be formed using one or more known processes. If the check result is negative, the circuit layout represented by the original graph 260 a is identified for revision.

FIG. 10 is a flow chart illustrating an n-pattern checking method 300, in accordance with some embodiments. The n-pattern checking method 300 is similar to the triple patterning checking method 100 a and the quadruple patterning checking method 200 a described above and similar description is not repeated herein. The n-pattern checking method 300 is configured to check a circuit layout for compatibility with an n-pattern lithography process that utilizes a predetermined number of photomasks (i.e., n photomasks) for patterning a circuit. For example, in various embodiments, n can equal any integer greater than 2.

At step 304, vertices having less than n edge connections are removed from the graph. Step 304 is similar to steps 104 of method 100 a and step 204 of method 200 b, but expanded to allow for the use of n colors in an n-pattern lithographic process. For example, in embodiments having n=5, step 304 removes any vertices having less than 5 (i.e. 4 or fewer) vertices. Similarly, in embodiment having n=6, step 304 removes any vertices having less than 6 (i.e., 5 or fewer) vertices.

If the method 300 is performing a first iteration, an initial process 306 is performed. On subsequent iterations through the method 300, the initial process 306 is skipped. During the initial process 306, each component in the graph (after removal of all vertices having less than n edges) are recorded at step 208. A check is performed at step 308 to determine whether any K_(n) graphs exist. If no K_(n) (or greater) graphs are identified, the method 300 proceeds to a K_(n−1) patterning reduction process 308. For examples, in embodiments having n=4, the K_(n−1) patterning reduction flow is a quadruple patterning reduction flow 200 a as described above. In some embodiments, the K_(n−1) reduction process 310 is an iteration of the n-pattern checking method 300 having n=n−1.

After performing the n-patterning reduction process 310, the method 300 checks the graph at step 316 to determine whether the graph is empty. If the graph is empty, the method proceeds to step 312. If the graph is not empty and/or the method 300 determines that the graph contains at least one K_(n) graph at step 308, the method 300 proceeds to step 310.

At least one K_((n+1)−1) reduction is performed at step 310. A K_(n+1) graph includes n+1 vertices each having an edge connection defining an outer perimeter and at least one inner edge connection. A K_((n+1)−1) reduction reduces the K_((n+1)) graph by combining a first vertex of the K_(n+1) graph with a second vertex of the K_(n+1) graph not directly connected to the first vertex. After performing the K_((n+1)−1) reduction, the method 300 can return to step 304.

In some embodiments, if the graph cannot be further reduced, either by a K_((n+1)−1) reduction, a K_(n−1) reduction process, and/or removal of vertices having fewer than n edges, the method 300 can proceed to step 312. At step 312, a n-patterning conflict check is performed. The n-patterning conflict check can include any suitable check, such as a brute-force n-color coloring, a heuristic n-patterning conflict check, a deterministic rules-based n-patterning conflict check, and/or any other suitable n-patterning conflict check. If the graph passes the n-patterning conflict check, the circuit layout can be formed by an n-photomask lithographic process. N photomasks, each corresponding to one of the n colors used in the n-patterning conflict check, can be generated for forming the circuit layout on a semiconductor substrate.

FIG. 11 illustrates one embodiment of a system 400 for generating a plurality of photomasks according to one or more embodiments of the methods disclosed herein. The system 400 includes at least one electronic device 402 configured to control operation of photomask generation system 404. U.S. Pat. No. 8,775,977, issued Jul. 8, 2014, entitled “Decomposition and Marking of Semiconductor Device Design Layout in Double Patterning Lithography” and U.S. Pat. No. 9,360,750, issued Jun. 7, 2016, entitled “Balancing Mask Loading” disclose photomask and semiconductor circuit generation systems and are incorporated by reference herein in their respective entireties. The electronic device 402 is capable of implementing one or more of the methods of generating a plurality of photomasks 100 a, 100 b, 200 a, 200 b, 300 described above. The photomask generation system 404 may generate a photomask according to one or more known methods. The electronic device 400 is a representative device and may comprise a processor subsystem 406, an input/output subsystem 408, a memory subsystem 410, a communications interface 412, and a system bus 414. In some embodiments, one or more than one of the electronic device 402 components may be combined or omitted such as, for example, not including the communications interface 412. In some embodiments, the electronic device 402 may comprise other components not combined or comprised in those shown in FIG. 11. For example, the electronic device 402 also may comprise a power subsystem. In other embodiments, the electronic device 402 may comprise several instances of the components shown in FIG. 11. For example, the electronic device 402 may comprise multiple memory subsystems 410. For the sake of conciseness and clarity, and not limitation, one of each of the components is shown in FIG. 11.

The processor subsystem 406 may comprise any processing circuitry operative to control the operations and performance of the electronic device 402. In various aspects, the processor subsystem 406 may be implemented as a general purpose processor, a chip multiprocessor (CMP), a dedicated processor, an embedded processor, a digital signal processor (DSP), a network processor, a media processor, an input/output (I/O) processor, a media access control (MAC) processor, a radio baseband processor, a co-processor, a microprocessor such as a complex instruction set computer (CISC) microprocessor, a reduced instruction set computing (RISC) microprocessor, and/or a very long instruction word (VLIW) microprocessor, or other processing device. The processor subsystem 406 also may be implemented by a controller, a microcontroller, an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a programmable logic device (PLD), and so forth.

In various aspects, the processor subsystem 406 may be arranged to run an operating system (OS) and various applications. Examples of an OS comprise, for example, operating systems generally known under the trade name of Apple OS, Microsoft Windows OS, Android OS, and any other proprietary or open source OS. Examples of applications comprise, for example, a telephone application, a camera (e.g., digital camera, video camera) application, a browser application, a multimedia player application, a gaming application, a messaging application (e.g., email, short message, multimedia), a viewer application, and so forth.

In some embodiments, the electronic device 402 may comprise a system bus 414 that couples various system components including the processing subsystem 406, the input/output subsystem 408, and the memory subsystem 410. The system bus 412 can be any of several types of bus structure(s) including a memory bus or memory controller, a peripheral bus or external bus, and/or a local bus using any variety of available bus architectures including, but not limited to, 9-bit bus, Industrial Standard Architecture (ISA), Micro-Channel Architecture (MSA), Extended ISA (EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB), Peripheral Component Interconnect Card International Association Bus (PCMCIA), Small Computers Interface (SCSI) or other proprietary bus, or any custom bus suitable for computing device applications.

In some embodiments, the input/output subsystem 408 may comprise any suitable mechanism or component to at least enable a user to provide input to the electronic device 402 and the electronic device 402 to provide output to the user. For example, the input/output subsystem 408 may comprise any suitable input mechanism, including but not limited to, a button, keypad, keyboard, click wheel, touch screen, or motion sensor. In some embodiments, the input/output subsystem 408 may comprise a capacitive sensing mechanism, or a multi-touch capacitive sensing mechanism.

In some embodiments, the input/output subsystem 408 may comprise a visual peripheral output device for providing a display visible to the user. For example, the visual peripheral output device may comprise a screen such as, for example, a Liquid Crystal Display (LCD) screen, incorporated into the electronic device 402. As another example, the visual peripheral output device may comprise a movable display or projecting system for providing a display of content on a surface remote from the electronic device 402. In some embodiments, the visual peripheral output device can comprise a coder/decoder, also known as a Codec, to convert digital media data into analog signals. For example, the visual peripheral output device may comprise video Codecs, audio Codecs, or any other suitable type of Codec.

The visual peripheral output device also may comprise display drivers, circuitry for driving display drivers, or both. The visual peripheral output device may be operative to display content under the direction of the processor subsystem 406. For example, the visual peripheral output device may be able to play media playback information, application screens for application implemented on the electronic device 402, information regarding ongoing communications operations, information regarding incoming communications requests, or device operation screens, to name only a few.

In some embodiments, the communications interface 412 may comprises any suitable hardware, software, or combination of hardware and software that is capable of coupling the electronic device 402 to one or more networks and/or additional devices (such as, for example, the photomask generating system 404.) The communications interface 412 may be arranged to operate with any suitable technique for controlling information signals using a desired set of communications protocols, services or operating procedures. The communications interface 412 may comprise the appropriate physical connectors to connect with a corresponding communications medium, whether wired or wireless.

Vehicles of communication comprise a network. In various aspects, the network may comprise local area networks (LAN) as well as wide area networks (WAN) including without limitation Internet, wired channels, wireless channels, communication devices including telephones, computers, wire, radio, optical or other electromagnetic channels, and combinations thereof, including other devices and/or components capable of/associated with communicating data. For example, the communication environments comprise in-body communications, various devices, and various modes of communications such as wireless communications, wired communications, and combinations of the same.

Wireless communication modes comprise any mode of communication between points (e.g., nodes) that utilize, at least in part, wireless technology including various protocols and combinations of protocols associated with wireless transmission, data, and devices. The points comprise, for example, wireless devices such as wireless headsets, audio and multimedia devices and equipment, such as audio players and multimedia players, telephones, including mobile telephones and cordless telephones, and computers and computer-related devices and components, such as printers, network-connected machinery such as a photomask generating system 404, and/or any other suitable device or third-party device.

Wired communication modes comprise any mode of communication between points that utilize wired technology including various protocols and combinations of protocols associated with wired transmission, data, and devices. The points comprise, for example, devices such as audio and multimedia devices and equipment, such as audio players and multimedia players, telephones, including mobile telephones and cordless telephones, and computers and computer-related devices and components, such as printers, network-connected machinery such as a photomask generating system 404, and/or any other suitable device or third-party device. In various implementations, the wired communication modules may communicate in accordance with a number of wired protocols. Examples of wired protocols may comprise Universal Serial Bus (USB) communication, RS-232, RS-422, RS-423, RS-485 serial protocols, FireWire, Ethernet, Fibre Channel, MIDI, ATA, Serial ATA, PCI Express, T-1 (and variants), Industry Standard Architecture (ISA) parallel communication, Small Computer System Interface (SCSI) communication, or Peripheral Component Interconnect (PCI) communication, to name only a few examples.

Accordingly, in various aspects, the communications interface 412 may comprise one or more interfaces such as, for example, a wireless communications interface, a wired communications interface, a network interface, a transmit interface, a receive interface, a media interface, a system interface, a component interface, a switching interface, a chip interface, a controller, and so forth. When implemented by a wireless device or within wireless system, for example, the communications interface 412 may comprise a wireless interface comprising one or more antennas, transmitters, receivers, transceivers, amplifiers, filters, control logic, and so forth.

In various aspects, the communications interface 412 may provide voice and/or data communications functionality in accordance a number of wireless protocols. Examples of wireless protocols may comprise various wireless local area network (WLAN) protocols, including the Institute of Electrical and Electronics Engineers (IEEE) 802.xx series of protocols, such as IEEE 802.11a/b/g/n, IEEE 802.16, IEEE 802.20, and so forth. Other examples of wireless protocols may comprise various wireless wide area network (WWAN) protocols, such as GSM cellular radiotelephone system protocols with GPRS, CDMA cellular radiotelephone communication systems with 1×RTT, EDGE systems, EV-DO systems, EV-DV systems, HSDPA systems, and so forth. Further examples of wireless protocols may comprise wireless personal area network (PAN) protocols, such as an Infrared protocol, a protocol from the Bluetooth Special Interest Group (SIG) series of protocols, including Bluetooth Specification versions v1.0, v1.1, v1.2, v2.0, v2.0 with Enhanced Data Rate (EDR), as well as one or more Bluetooth Profiles, and so forth. Yet another example of wireless protocols may comprise near-field communication techniques and protocols, such as electro-magnetic induction (EMI) techniques. An example of EMI techniques may comprise passive or active radio-frequency identification (RFID) protocols and devices. Other suitable protocols may comprise Ultra Wide Band (UWB), Digital Office (DO), Digital Home, Trusted Platform Module (TPM), ZigBee, and so forth.

In some embodiments, at least one non-transitory computer-readable storage medium is provided having computer-executable instructions embodied thereon, wherein, when executed by at least one processor, the computer-executable instructions cause the at least one processor to perform embodiments of the methods described herein. This computer-readable storage medium can be embodied in memory subsystem 410.

In some embodiments, the memory subsystem 410 may comprise any machine-readable or computer-readable media capable of storing data, including both volatile/non-volatile memory and removable/non-removable memory. The memory subsystem 410 may comprise at least one non-volatile memory unit. The non-volatile memory unit is capable of storing one or more software programs. The software programs may contain, for example, applications, user data, device data, and/or configuration data, or combinations therefore, to name only a few. The software programs may contain instructions executable by the various components of the electronic device 402.

In various aspects, the memory subsystem 410 may comprise any machine-readable or computer-readable media capable of storing data, including both volatile/non-volatile memory and removable/non-removable memory. For example, memory may comprise read-only memory (ROM), random-access memory (RAM), dynamic RAM (DRAM), Double-Data-Rate DRAM (DDR-RAM), synchronous DRAM (SDRAM), static RAM (SRAM), programmable ROM (PROM), erasable programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), flash memory (e.g., NOR or NAND flash memory), content addressable memory (CAM), polymer memory (e.g., ferroelectric polymer memory), phase-change memory (e.g., ovonic memory), ferroelectric memory, silicon-oxide-nitride-oxide-silicon (SONOS) memory, disk memory (e.g., floppy disk, hard drive, optical disk, magnetic disk), or card (e.g., magnetic card, optical card), or any other type of media suitable for storing information.

In one embodiment, the memory subsystem 410 may contain an instruction set, in the form of a file for executing a method of generating one or more graphs (for example, from one or more circuit layouts provided to the electronic device 402), reducing the one or more graphs and checking the one or more graphs for compatibility with one or more photolithographic processes, as described herein. In some embodiments, the memory subsystem 410 contains instructions for optionally generating a plurality of photomasks 100 a, 100 b, 200, 200 b, 300 using the photomask generating system 404. The instruction set may be stored in any acceptable form of machine readable instructions, including source code or various appropriate programming languages. Some examples of programming languages that may be used to store the instruction set comprise, but are not limited to: Java, C, C++, C#, Python, Objective-C, Visual Basic, or .NET programming. In some embodiments a compiler or interpreter is comprised to convert the instruction set into machine executable code for execution by the processing subsystem 406.

It is understood that that the above described arrangements are only illustrative of the application of the principles of the present disclosure. Numerous modification and alternative arrangements may be devised by those skilled in the art without departing from the spirit and scope of the present invention. Thus, while the present invention has been shown in the drawing and fully described above with particularity and detail in connection with what is presently deem to be the practical and preferred embodiments of the disclosure, it will be apparent to those of ordinary skill in the art that numerous modifications, including, but not limited to, variations in size, materials, shape, form, function, manner of operation, assembly, and use may be made without departing from the principles and concepts of the disclosure.

In various embodiments, a method of generating a plurality of photomasks for a photolithographic process includes generating a circuit graph representative of a circuit layout having a plurality of conductive lines. The graph comprises a plurality of vertices and a plurality of edges. Each of the plurality of vertices is representative of a corresponding one of the plurality of conductive lines and each of the plurality of edges is representative of a spacing between the conductive lines less than an acceptable minimum distance. At least one Kn+1 graph is identified within the circuit graph. A Kn+1 graph comprises a first set of vertices selected from the plurality of vertices connected in series by a first set of edges selected from the plurality of edges and having at least one non-series edge connection between a first vertex and a second vertex selected from the first set of vertices. The first set of vertices comprises n+1 vertices, where n is any integer greater than 2. The at least one Kn+1 graph is reduced by merging a third vertex into a fourth vertex selected from the first set of the plurality of vertices. The third vertex has a second set of edges selected from the plurality of edges and the fourth vertex has a third set of edges selected from the plurality of edges. The third vertex is merged into the fourth vertex by removing the third vertex from the circuit graph and adding edges from the second set of edges to the third set of edges to generate a reduced circuit graph. The edges from the second set of edges are added to the third set of edges only if a the third set of edges does not include a corresponding edge. An n-pattern conflict check is performed on the reduced circuit graph. The plurality of photomasks are generated based on a result of the n-pattern conflict check.

In various embodiments, a method includes receiving a circuit layout comprising a plurality of conductive lines and generating a circuit graph representative of the circuit layout. The circuit graph comprises a plurality of vertices and a plurality of edges. Each of the plurality of vertices represents one of the plurality of conductive lines. Each of the plurality of edges is representative of a spacing between the conductive lines less than an acceptable minimum distance. The method further includes coloring the circuit graph with n colors where n is equal to any integer greater than 3. Each of the plurality of vertices is assigned one of the n colors based on an n-patterning checking process including at least one K(n+1)−1 graph reduction. The method further includes assigning each of the plurality of conductive lines to one of n photomasks based on a color of a corresponding vertex from the plurality of vertices and forming the n photomasks for use in a n-patterning photolithographic process.

In various embodiments, a method of generating a plurality of photomasks for a photolithographic process includes generating a circuit graph representative of a circuit layout having a plurality of conductive lines. The circuit graph comprises a plurality of vertices and a plurality of edges. Each of the plurality of vertices is representative of one of the plurality of conductive lines and each of the plurality of edges is representative of a spacing between the conductive lines less than an acceptable minimum distance. The circuit graph is reduced by one or more K₄₋₁ reductions and one or more square loop reductions to generate a reduced circuit graph comprising a set of the plurality of vertices in the circuit graph. The reduced circuit graph is checked for one or more triple patterning conflicts. The checking assigns one of three colors to each of the set of vertices in the reduced circuit graph. A first photomask, a second photomask, and a third photomask each comprising a set of the plurality of conductive lines of the circuit layout are generated. Each set of the plurality of conductive lines corresponds to one of three colors assigned to the set of vertices. The first color corresponds to the first photomask, the second color corresponds to the second photomask, and the third color corresponds to the third photomask.

The foregoing outlines features of several embodiments so that those skilled in the art may better understand the aspects of the present disclosure. Those skilled in the art should appreciate that they may readily use the present disclosure as a basis for designing or modifying other processes and structures for carrying out the same purposes and/or achieving the same advantages of the embodiments introduced herein. Those skilled in the art should also realize that such equivalent constructions do not depart from the spirit and scope of the present disclosure, and that they may make various changes, substitutions, and alterations herein without departing from the spirit and scope of the present disclosure. 

What is claimed is:
 1. A method of generating a plurality of photomasks for a photolithographic process, comprising: generating a circuit graph representative of a circuit layout having a plurality of conductive lines, wherein the graph comprises a plurality of vertices and a plurality of edges, wherein each of the plurality of vertices is representative of a corresponding one of the plurality of conductive lines, and wherein each of the plurality of edges is representative of a spacing between the conductive lines less than an acceptable minimum distance; performing at least one K_(n+1) graph reduction of the circuit graph to generate a reduced circuit graph; performing an n-pattern conflict check on the reduced circuit graph; and generating a plurality of photomasks based on a result of the n-pattern conflict check.
 2. The method of claim 1, comprising removing from the circuit graph a second set of vertices selected from the plurality of vertices, wherein each of the vertices in the second set of vertices has less than n edge connections.
 3. The method of claim 1, comprising recording each of the plurality of vertices and each of the plurality of edges prior to performing the at least one K_(n+1) graph reduction.
 4. The method of claim 1, wherein prior to performing at least one K_(n+1) graph, the method comprises: identifying at least one K_(n) graph within the circuit graph, wherein a K_(n) graph comprises a second set of vertices selected from the plurality of vertices connected in series by a second set of edges selected from the plurality of edges and having at least one non-series edge connection between a first vertex and a second vertex selected from the second set of vertices, wherein the second set of vertices comprises n vertices; and merging a third vertex selected from the second set of vertices into a fourth vertex selected from the second set of vertices, wherein the third vertex has a vertex-specific set of edges selected from the second set of edges, and wherein the third vertex is merged into the fourth vertex by removing the third vertex from the circuit graph and adding the vertex-specific set of edges of the third vertex to the fourth vertex to generate the reduced circuit graph.
 5. The method of claim 1, comprising checking the plurality of edges for one or more edges including at least one of a loop or an empty edge.
 6. The method of claim 1, wherein the n-pattern conflict check is selected from the group consisting of: a brute-force n-color check, a heuristic n-pattern conflict check, and a deterministic rules-based conflict check.
 7. The method of claim 1, wherein the Kn+1 graph reduction comprises a square loop reduction process.
 8. The method of claim 1, comprising iteratively identifying one or more additional K_(n+1) graphs and reducing the one or more additional K_(n+1) graphs.
 9. The method of claim 1, wherein n is equal to three, and wherein the at least one K_(n+1) graph is a K₄ graph.
 10. The method of claim 1, wherein n is equal to two, and wherein the at least one K_(n+1) graph is a K₃ graph.
 11. A method, comprising: receiving a circuit layout comprising a plurality of conductive lines; generating a circuit graph representative of the circuit layout, wherein the circuit graph comprises a plurality of vertices and a plurality of edges, wherein each of the plurality of vertices represents one of the plurality of conductive lines, and wherein each of the plurality of edges is representative of a spacing between the conductive lines less than an acceptable minimum distance; coloring the circuit graph with n colors where n is equal to any integer greater than 3, and wherein each of the plurality of vertices is assigned one of the n colors based on an n-patterning checking process including at least one K_((n+1)) graph reduction; assigning each of the plurality of conductive lines to one of n photomasks based on a color of a corresponding vertex from the plurality of vertices; and forming the n photomasks for use in a n-patterning photolithographic process.
 12. The method of claim 11, wherein the K_((n+1)) graph reduction comprises removing a set of vertices selected from the plurality of vertices from the circuit graph, wherein each of the set of vertices has less than n edge connections.
 13. The method of claim 11, wherein the n-pattern checking process comprises an n-pattern conflict check selected from the group consisting of: a brute-force n-color coloring, a heuristic n-pattern conflict check, and a deterministic rule-based conflict check.
 14. The method of claim 11, where in the n-pattern checking process comprises at least one square loop reduction.
 15. The method of claim 11, comprising iteratively identifying one or more additional K_(n+1) graphs and reducing the one or more additional K_(n+1) graphs.
 16. The method of claim 11, wherein n is an integer greater than or equal to
 3. 17. A system configured to generate a plurality of photomasks, the system comprising: a computer configured to: receive a circuit layout comprising a plurality of conductive lines; generate a circuit graph representative of a circuit layout having a plurality of conductive lines, wherein the circuit graph comprises a plurality of vertices and a plurality of edges, wherein each of the plurality of vertices is representative of one of the plurality of conductive lines, and wherein each of the plurality of edges is representative of a spacing between the conductive lines less than an acceptable minimum distance; reduce the circuit graph by one or more K_(n+1) reductions to generate a reduced circuit graph; and check the reduced circuit graph for one or more triple patterning conflicts, wherein the checking assigns one of three colors to each of the set of vertices in the reduced circuit graph; and a photomask generator configured to generate a first photomask, a second photomask, and a third photomask each comprising a set of the plurality of conductive lines of the circuit layout, wherein each set of the plurality of conductive lines corresponds to one of three colors assigned to the set of vertices, wherein the first color corresponds to the first photomask, the second color corresponds to the second photomask, and the third color corresponds to the third photomask.
 18. The system of claim 17, wherein the computer is configured to remove one or more vertices from the plurality of vertices, wherein the one or more vertices each have less than three edges associated therewith in the plurality of edges.
 19. The system of claim 17, wherein the one or more K_(n+1) reductions comprises merging a first vertex into a second vertex selected from a first set of the plurality of vertices defining a K_(n) loop, wherein the first vertex has a set of edges selected from the plurality of edges, and wherein the first vertex is merged into the second vertex by removing the first vertex from the circuit graph and adding the set of edges to the second vertex to generate a reduced circuit graph.
 20. The system of claim 19, wherein the first vertex and the second vertex have a largest path separation of pairs of vertices from the first set of the plurality of vertices. 